Although (REF ) approximates the \(\ell _0\) -norm, we note that \(\left\Vert \mathbf {x}_{j,1}+\epsilon \mathbf {1}\right\Vert _q^q\)  is concave and (REF ) is not easy to minimize. Therefore, if we add \(c_a\left\Vert \mathbf {x}_{j,1}+\epsilon \mathbf {1}\right\Vert _q^q\)  to the objective function of (REF ), it is still hard to solve. To tackle this problem, we consider a quadratic upper-bound [1]} for (REF ) and successively minimize the upper-bound [2]}. We denote the \(i^{\text{th}}\)  iterate of \(\mathbf {x}_{j,1}\)  by \(\underline{\mathbf {x}}_{j,1}^i\)  and obtain the following upper-bound:
\(&\left\Vert \mathbf {x}_{j,1}+\epsilon \mathbf {1}\right\Vert _q^q \le \left\Vert \underline{\mathbf {x}}_{j,1}^i+\epsilon \mathbf {1}\right\Vert _q^q\nonumber \\&+q\big \langle \lbrace (\underline{x}_{s,1}^i+\epsilon )^{q-1}\rbrace _{s\in \mathcal {S}_j}, (\mathbf {x}_{j,1}-\underline{\mathbf {x}}_{j,1}^i)\big \rangle +\frac{e}{2}\left\Vert \mathbf {x}_{j,1}-\underline{\mathbf {x}}_{j,1}^i\right\Vert _2^2,\) 
